# thq_05 - O A05/015/01 0 Use T.C 16s MATH 2451 thq_05 Mult...

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Unformatted text preview: O. A05/015/01 0 Use T.C. 16s MATH 2451' thq_05 Mult Exam ble Calculus 1var1a Course Due Date 2016-o2~o4 (Thu): McCary Instructor . Net ID e W m 0 AI @ @ ® @ ewe [email protected] ® @ «my O @ me E, as . _ . ® ® 3 . [email protected]@@©©@@®@@@@©eaeeweeeee®ee®eeeeaa [email protected]@@@®@®@C®®®seemsas Ix 1.. _ 3 «ma ® Ame [email protected] weeeeeeeeweeee my 0 ® @ E ® @ Amy © me ® @ © Ame /i ﬂee eﬁeeeeeeeee®ea m ‘\l ‘ .I II my E O ® ® ® ® .. , .l 3 ® ® ® ® ® ® @ @ ma ® © @ ® @ ® L @ ® ® @ @ aw @ mt, «UV m w @ ﬂ. @ ® ® (First Name) xU/ a“ .. raw A,..\ Q7/1986 ®@@@@@®®@ saeeeea k )1 me, Q @H (Last Name) 07 - 2018-01-28 14 .. A05/015/02 . 19112::vn2- 1. (10 points) Find all values of a. and b which make the function differentiable zit all points. You must use left ane limits= the limit deﬁnitibn of continuity at a. point, and the limit deﬁnition of derivative at a point. PGF‘WN I110"! meeég A’T—D— . éamﬁ'é 5r: Iﬁrl up; 1*0121‘22 ﬁT§§;?Jz§JT";4véé— f() am2+b\$+4 I .LI‘JT-C 'i' m = ‘-—’ - a\$+b 2:22,! a, ﬁ<”>j;_ FM? vﬂpmé‘; as AWL" Fm- iﬁég} é 90M;.—-_iv.g,d_; | W39 a"); ﬁm-Jrlbﬂt (Lew? 2* 4a L; M 73):; 2am Mtge}: "' 'r' ' V» a 5.2. N 9157‘) léL/JNHWM'C’l/LS W V/Y' plF‘F‘éﬂ/‘efj? ’f’kréi'f WWW emf—iaihﬁrrj/xr b - ' ' 4.6-1? : 2A -% L; —7 #4} H _' f5“ 1 I__ h / )0 '1 Lari M:- EA 1'. 7—? "’ 1/; I-ﬁi—‘g i‘;':,.:-“;. DN/ Irv aﬂr‘fﬁ F3»? '-FL":‘~'1'_,} T5 ‘33 _'2"’«: '41:- . 5.1-». in z: -:_-.~r:v°c;a,-:/ Km (49' . ﬂy “CW? WWW-9 feméAwp ll. LAM-HE; Mmﬁ MATCH- A1— 15; i :- _ a» \ . ' '[email protected]—/ ‘4 '1 H— 11. .. 6012’; putt-*gz-ig’N’IM-BLE’ Afar; 1 l9 1 '- l2";— FF: iGraderOnlyJr O 069 o O. A05/015/OL O _;-0 points) n . ,— n , . (a) Compute frag-gm) using the limit deﬁnition of partial derivative. 1 lie-i L7,: (b) Compute using the limit deﬁnition of partial derivative, "7 In 3 \ r F r ‘ (c) Show f is not continuous at (l. ,. ‘f- l- ,‘“ y 0 otherwisezﬁuéql “A? w W. ‘ ’Y 1;": ih};_..., # _ E. l is)? gaff ages {are 117 + 11'») 5 (>554 7: reef—- ﬁg, f K lﬂv‘ﬁ 0 "' (a)?! ,x—y \. v 965 "N w in“, C ﬁDMIrvl/LH ‘lmy'aLté‘Si 1i“ K t ht game h’l/l ﬁll i 42(0):?) lt wrap If Dir-7“] f i I ‘ I “265) r -%Q" :15 HPLQIQET’U :‘0 i If l" PJ lg NOT ”' (GN'ilL‘VL'1‘45’i‘: "g1 g Q “ \ ' n - a ‘ ‘ V“ f ' :7“ . 7 1, __ / V - i“ _ .* ti! A ._ .4 ‘ - a *U " “477171.41?- ' ‘- ‘ J J .1 %, \3'"Lll-'lr'\hlzc'w L M"- -. J/lW’l Qt: T: 0 M” “’6‘ ’ ' A " 1 ‘ 5' w, " ~——/\_._——-——~ The purgste of this exercise is to illustrate one of the weaknesses of partial derivatives: even if all partials exist at a, point, the function may still be discontinuous! This is in contrast to scalar calculus Where the existence of g’(a) implies continuity at a. See the Comment about symbolic calculators at the end of the next exercise, which also holds for this exercise. (ﬂ - ‘k .1 J, Grader Only J, 6 Pea-2n £4, 9‘ ‘ o o A o 5 / 0 1 5 / 0,4 9 {£99549 ores-a2 W” WA ‘H 17 C?” rape”, F _. ‘_ ~ ‘ ‘ 'nﬁf cgeﬁnitio‘ff oi‘partial derivative. / I j _ '_.- 1“ 'u'n -‘ Ir'm'il I .- ‘ - i" "/43 7H ' ' I. I my :1: y y / V' .‘ m #5 inf/.7" / ./ f 2 3 +9 / 0 otherwise 3. (10 points) Find 3—56), \$503), \$1346), and gays) usingith— ’ r c, -. r .. " ,3 ’f —— 7—\ \7 '3- . «If " W's) (WW/F" ’7' 92—» “’1 . :" 74 7 add“ ‘1‘ j ——-'>< Pa”- ..I-ix’“ in iv ; ‘7' £1330 Hurts‘LéZQhW-f} ' "ﬁzzﬁwf fax? 7 _ d Z I...‘ ‘ J “I: _ I E :3 is}? I \ z: ' [4375’] «HA? 3+9. 7—75? 3]“: a an 4' films? 2'; W V; H“ Jaf 4 4 2:. ,5 y, :12} 4 f g} Liege - m1— e—Y‘aii‘n—“v ‘2’ lf‘erm a . roe/+21% page — 7‘anch 143771- 2137 Baldw- ”+-an+121,1~,? "l—L 4- ; 2 7— 4 r ﬂ -} 3 3:7 7 a r” A 5 *2"? 7’“ “H4 ‘47—”‘7‘jl W “1/7" itiygﬂi’sflﬂwiﬁ)’ — g 7 3 3 it}??? 411:“; 3 7 +1.? 3‘" xii/75%“ r,” 1m: 7 a» L (“A ,{7 asleewesna_, W”) L" "Q “irreftyz’l We ﬁgﬁtésﬁluéh%a w 2 . W 5 ﬂush-ﬁgure)‘er—W gran-t - ‘L‘m’tr‘f’ri‘i’ at (WW/i); (“ﬂﬁﬂﬂmeawr 14 mph/7):, #0 i {94" 9‘15? . . . . . . ” ./ I ‘ . . The purpose of th1s exerc1se is to show that mixed partlals 1n dgi'grentporders are not always equal- Don‘t get lost 1n the limit formalism — try pier-ting assisﬁﬁa second partials onﬁlﬁ‘am Alpha (or, even better, download SAGE Math and start using it, MATH majors). Can you see why the mixed partials are not equal? Feel free to use a symbolic calculator to compute the partials of f away from 5, but remember that you must use the limit deﬁnition to compute the partials at 6 because 5 is not in the domain of that rational expression. l Grader Only J, . O® @©@@@@©®@@® . O. A05/OlS/05 0 ,0 points) Compute the Jacobian. You must Show your work and not use a symbolic calculator. I. (all in -' J 33;? r énvoéf (3 "._ l 7." i 2-way “-‘ ’ ' z / . lQX‘r/s '; 1 '60?" * I l /r ’ ‘__ f [ J, Grader Only J, .. A05/015/06 . 5. (10 points) Let f = Show, using limits, that for any number m, 5%(i(o+h)—r(0)—mh)=0 3;}: 3x23 (” VA?“ 'r-J Q r>< but f1}i_rrb%(f(0+h)—f(0)—mh)=0 is never true: there is no m such that mh is a good approximation to the increment A f at U. (rgfo+-t)-£(5)m)j )om mt mime/mt e i. m. t w— My \‘F \A «-----><', Eli-1X u we in ‘. r W-Ji‘h Mil/i) : p/M’W "IA" m m z- Wm) :- O [/1':?OH‘ lﬂﬁr 0:! , lﬂaoﬁ ‘ LlMl'r' Ewe—rg__,«i{r><- la; CﬂN—WNMGMS Aron _ M 7w» \/ WWW» Mt :0 [/80 f {£{Ui Life 1? ‘3‘}- Iii”; i: -},V,lril/El Z“: ,'_‘/V‘1 W1 \ﬁﬂPQH/l/‘r Maj—min )Jim fog/1N hm) = -I _ . "m I” rm UM” VINE hog/f .- f, w 4m H'Q\/(/THé AEEO Luff}? V/XLMC TbtpV"G| loN l5 00mm 0m” AT 0, E: :T’ 1:; N55” mesa/126m mate The purpose of this exercise is to wean you off of the scalar calculus deﬁnition of derivative and to get you accustomed to the Fréohet deﬁnition- iGraderOnlyJ, . O® @@@®@©©®®®Q . ...
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